91Ó°ÊÓ

(a) Use a graphing utility to confirm that the graph of \(r=2-\sin (\theta / 2)(0 \leq \theta \leq 4 \pi)\) is symmetric about the \(x\) -axis. (b) Show that replacing \(\theta\) by \(-\theta\) in the polar equation \(r=2-\sin (\theta / 2)\) does not produce an equivalent equation. Why does this not contradict the symmetry demonstrated in part (a)?

Use a graphing utility to investigate how the family of polar curves \(r=1+a \cos n \theta\) is affected by changing the values of \(a\) and \(n\), where \(a\) is a positive real number and \(n\) is a positive integer. Write a brief paragraph to explain your conclusions.

If \(f^{\prime}(t)\) and \(g^{\prime}(t)\) are continuous functions, and if no segment of the curve $$ x=f(t), \quad y=g(t) \quad(a \leq t \leq b) $$ is traced more than once, then it can be shown that the area of the surface generated by revolving this curve about the \(x\) -axis is $$ S=\int_{a}^{b} 2 \pi y \sqrt{\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2}} d t $$ and the area of the surface generated by revolving the curve about the \(y\) -axis is $$ S=\int_{a}^{b} 2 \pi x \sqrt{\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2}} d t $$ [The derivations are similar to those used to obtain Formulas (4) and (5) in Section 6.5.] Use the formulas above in these exercises. The equations \(x=a \phi-a \sin \phi, \quad y=a-a \cos \phi \quad(0 \leq \phi \leq 2 \pi)\) represent one arch of a cycloid. Show that the surface area generated by revolving this curve about the \(x\) -axis is given by \(S=64 \pi a^{2} / 3\)

Why do you think the adjective "polar" was chosen in the name "polar coordinates"?

What are some of the advantages of expressing a curve parametrically rather than in the form \(y=f(x)\) ?